Well, based on my experience (255 Hours of Gameplay) Skyrim those above, seem to be true for me, with every character.
Stealth Assassination Finishing Moves
- Happens most of the time when the target is a human and is standing up. Any direction will work here, from sides, behind or even in the belly.
- Will only happen if you have enough damage to one hit kill the target on your sneak atack. If your damage is not able to one shot the enemy, no animation will be shown, just a normal bonus sneak atack. As it is a finishing move, the enemy won't stay alive after the hit.
Close Ranged Finishing Moves ( Both One handed and Two Handed Weapons )
Holding the sprint button and running towards a enemy from a close range (Not running the entire map, it has to be close) and power atacking happens to proc the animation more offen. Even without perks in those trees.
Again, the damage must be high enough to kill the enemy. As it is a finishing move, the enemy won't stay alive after the hit.
More Skills or Critical Strike Chance Mean More Finishing Moves:
Since the damage must be high enough to kill the target for the finishing move to happen, if you have a higher one handed skill level for instance, your damage output will be higher, allowing you to execute enemies more frequently.
Same applies for critical, since critical helps the damage output, with a higher damage, you will be able to kill enemies that you wouldn't be able to without a critical strike.
Needs Testing Here:
I am not sure if the enchantings and potions bonuses apply after or before the sneak atack multiplier. But if the damage calculated for the multiplier involves the extra damage from enchantings and potions, it would also help you to get more animations.